THERMOELASTIC EFFECTS IN VITREOUS SILICA

HAL Id: jpa-00220852

https://hal.archives-ouvertes.fr/jpa-00220852

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THERMOELASTIC EFFECTS IN VITREOUS SILICA

O. Wright, W. Phillips

To cite this version:

O. Wright, W. Phillips. THERMOELASTIC EFFECTS IN VITREOUS SILICA. Journal de Physique

Colloques, 1981, 42 (C4), pp.C4-1017-C4-1020. �10.1051/jphyscol:19814222�. �jpa-00220852�

JOURNAL DE PHYSIQUE

CoZZoque C4, suppZ6ment au nO1O, Tome 42, o c t o b r e 1981 page C4-1017

THERMOELASTIC EFFECTS I N VITREOUS S I L I C A

O.B. Wright and W.A. Phillips Cavendish Laboratory, Cambridge, U. K.

Abstract.- The ~ G n e i s e n parameter o f v i t r e o u s s i l i c a i s o f o r d e r -10 and s t r o n g l y temperature dependent a t 1.5 K, whereas i n most c r y s t a l l i n e s o l i d s i t i s o f o r d e r +l and temperature independent a t s i m i l a r temperatures. I n t h i s i n v e s t i g a t i o n t h e G r i n e i s e n parameter has been determined by means o f t h e r m o e l a s t i c measurements. The temoerature changes i n rods o f v i t r e o u s s i l i c a were monitored when mechanical s t r a i n s were a p p l i e d i n t h e tempera- t u r e ranqe 1.5

-

17 K. Below -5 K t h e Grfineisen parameter was found t o depend on t h e average s t a i n o f t h e sample; t h i s dependence became more marked as t h e temperature was lowered, a n d a t d . 5 K t h e Griineisen parameter became p o s i t i v e f o r l a r g e average s t a i n s . I r r e v e r s i b l e e f f e c t s were a l s o observed i n d i c a t i n g a broad d i s t r i b u t i o n o f r e l a x a t i o n times extending t o times longer than 100s.

The t e r n ' t h e r m o e l a s t i c e f f e c t ' describes t h e a d i a b a t i c temperature change

&T produced by a change i n volume 6 V . For a r e v e r s i b l e change

-

^- _{- -y} ^&

v ,

T ( 1

where y i s t h e G r i n e i s e n parameter, T h i s parameter has p r e v i o u s l y been determined i n v i t r e o u s s i l i c a a t temperatures above 1.5 K u s i n g thermal expansion data (1, 2).

I n t h i s i n v e s t i g a t i o n , mechanical s t r a i n s were a p p l i e d t o samoles o f v i t r e o u s s i l i c a a t low temperatures, a l l o w i n g y t o be determined d i r e c t l y u s i n g equation 1 (see e.g. von Schickfus, H., Diplomarbeit, Jan. 1974, unpublished).

Measurements were made i n t h e temperature range 1.5

-

17 K. The samples were rods o f S p e c t r o s i l and \ ! i t r e o s i l (obtained from Thermal Syndicate Ltd.) o f dimen- sions l mm X 29 cm, and were attached a t b o t h ends t o g r i p s mounted i n an

evacuated can. L o n g i t u d i n a l s t r a i n s ranging from 10-5 t o 5 X 10-" were used. The temperature a t t h e c e n t r e o f t h e sample was measured u s i n g a carbon r e s i s t a n c e thermometer designed t o have a low h e a t capacity.

I n t h e f i r s t type o f experiment performed, ' s t e p - f u n c t i o n ' s t r a i n s viere a p p l i e d (as shown i n t h e i n s e t o f Fig. 1). S t r e t c h i n o times a t i n t h e range 0.01 < a t < 10 ^S were used. Fig. 1 shows t h e r e s u l t s o b t a i n e d f o r V i t r e o s i l a t 4.7 K. The sample temperature decays back t o i t s i n i t i a l v a l u e because o f thermal conduction w i t h p r i n c i p a l time constant ^{T O} = L2cp/r2r, where L i s t h e sample length, c t h e s p e c i f i c h e a t capacity, r, t h e d e n s i t y , and K t h e thermal c o n d u c t i v i t y ;

(7, % 20 S a t 4.7 K ) . The temperature chanoes a r e n o t r e v e r s i b l e b u t can be analysed i n t o symmetric and antisymmetric c o n t r i b u t i o n s . Fig. l c shows t h e a n t i - symmetric component o f temperature change which i s obtained by t a k i n g h a l f t h e d i f - ference between t h e t r a c e s i n F i g . l a and l b . Provided a t << T

,

t h e temperature change i s approximately a d i a b a t i c f o r times t & 9 . 2 ~ ~ ~ and t h e t r ~ n e i s e n parameter can be c a l c u l a t e d from t h e maximun asymmetric temperature change ATa u s i n g

equation 1 (where 6V/V = ( l

-

? v ) E ~ , V being Poisson's r a t i o ) . For temperatures

> 4 K AT, was found t o be p r o p o r t i o n a l t o t h e s t r a i n change ^.E, F u r t h e r , ATa d i d n o t depend on ~t p r o v i d e d ~t

i

0 . 2 ~ ~ . By c o n t r a s t , t h e maximum symmetric

temperature change dTs ( o b t a i n e d from the average o f Fi?. l a and l b

-

see Fig. I d )

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19814222

C4-1018 JOURNAL DE PHYSIQUE

F i g u r e 1: Experimental t r a c e s o f temperature a g a i n s t time obtained a t T = 4.7 K f o r

~t = 0.2 s and E, = 3.1 X 10-4. Traces a ) and b) correspond t o r e s p e c t i v e l y s t r e t c h i n g and u n s t r e t c h i n g t h e sample. Curves c ) and d ) a r e r e s p e c t i v e l y t h e smoothed antisymmetric and symmetric temnerature changes. The i n s e t shows t h e a p p l i e d s t r a i n as a f u n c t i o n o f time.

was found t o be p r o p o r t i o n a l t o em2 over the whole temperature range. I n t h e a d i a b a t i c regime ( ~ t

2

0.2 T,), Us decreased l o g a r i t h n i c a l l y as a t was increased.

Below % 4 K AT, became t o o l a r g e t o a l l o w an accurate d e t e r m i n a t i o n o f ^y.

I n t h e second t y p e o f experiment, s i n u s o i d a l s t r a i n s o f t h e form

were applied, where ~ ( t ) i s t h e time v a r y i n n l o n q i t u d i n a l s t r a i n , eav t h e averaae s t r a i n , and EO t h e s t r a i n amplitude. 9n a p p l y i n g t h i s s t r a i n t o the sample, t h e average temperature increased by an amount AT, t o a steady s t a t e value. The temperature change AT, i s r e l a t e d t o t h e i n t e r n a l f r i c t i o n Q - I through t h e r e l a t i o n

1 ~KAT,

Q-l =

W-

^{( 3 )}

where Y i s t h e Young's modulus of t h e sample. For t h e exnerimental frequency range 0.005

-

50 Hz, &?-l was found t o be indenendent o f b o t h ^OJ and ^E. For t h e \ l i t r e o s i l sample, &-l increased smoothly from 3 X 10-4 a t 4 K t o 5 X a t 10 K uhereas f o r S p e c t r o s i l i t increased from 3 X a t 4 K t o 5 X 10-4 a t 17 K. Relo1.1 Q K, Q-1 was temperature indenendent i n b o t h cases.

Superimposed on t h e steady s t a t e temperature r i s e was an o s c i l l a t i n o temnera- t u r e component. The temperature v a r i a t i o n 6 T ( t ) f o r both s i l i c a s c o u l d be

modelled as t h e sun o f a fundamental and second harmonic component according t o t h e equation

provided t h a t t h e frequency was h i q h enough (UT, $ 5) f o r t h e t e v p e r a t u r e v a r i a t i o n t o be approximately a d i a b a t i c . The two components c o u l d be detected s e p a r a t e l y and w i t h i n t h e experimental accuracy b o t h AT1 and AT, were found t o be frequency

independent i n t h e a d i a b a t i c reoime. !.S t h e temperature was raised, AT, decreased

-101 I I I

J

0 1 2 3 L

average straln X 10L

Fiqure 2: Plot of

y

against

E,,

with

E O = 2 . 5 X

10-5 a t three temoeratures:

a ) 1.7

K

with

^U =

3.4 rad

S - ) ;

b) 2.8

K

with

^W = 6.3

rad

S - l ;

c )

3.6

K with

= 6.8

rad

S-l.

The dotted l i n e s indicate extrapolation.

-20 0 5 10

temperature In K

Figure 3: Plot of GrUneisen narameter aoainst temoerature. Crosses and c i r c l e s a r e t h e r e s u l t s of t h i s exnerinent f o r Vi t r e o s i l and Spectrosil resnectively.

Curve a ) i s calculated from t h e r e s u l t s of Lyon e t a l . ( 2 ) f o r Spectrosil and curves

h)

and c ) from the r e s u l t s of I*lhite (1

)

f o r \ l i t r e o s i l and Snectrosil resnectively.

and above

^Q, 5 K

i t could not be detected. I t was found t h a t

AT,

was pro?ortional t o and independent of

E,,.

The Grineisen naraveter as defined by equation

1

was calculated from AT^. Above

^Q,

5

K y

was independent of s t r a i n and agreed

w i t h

the r e s u l t s of the 'step-function' measurements; below

^%

5

K

i t varied l i n e a r l y

w i t h

the average s t r a i n as shown i n Fiq.

2.

For both s i l i c a s

a y / a e a V

was pronortional t o

1 / ~ 3 .

A t 1.6

K

a Y / a ~ a V was larqe

(Q, 5 X

l q 4 ) and

y

became nositive f o r cav

>

2.5 X

lO-'+. The ~ r i n e i s e n parameter f o r zero s t r a i n v(€=')) bras obtained by extra- polation as shown in Fiq. 2. Fiqure

3

shows a n l o t of aqainst

T,

together with t h e Grineisen parameter calculated from thermal expansion measurements.

Although the estimated accuracy of the present measurements i s only

20%,

t h e rapid f a l l - o f f of

y

observed i n both s i l i c a s be'lotd

^,L

3

K

appears t o aqree b e t t e r with the r e s u l t s of klhite than those of Lyon e t a l .

Previous internal f r i c t i o n measurements i n vitreous s i l i c a have been made i n the frequency range

5 kHz

-

³⁰ GHz using ultrasonic techniques. A t

temperatures above 10 K these e f f e c t s have been explained usin? a c l a s s i c a l two-level system model of a n e l a s t i c relaxation

( 3 ) .

If the r e s u l t s of these measurements are extra- polated t o frequencies of order 1 Hz,both the magnitude and temperature dependence found f o r

&-l

a r e consistent with the present r e s u l t s . A t low temneratures,

&-l

should tend t o t h e l i m i t civen by quantum-mechanical theory. The internal f r i c t i o n a t frequencies of order 1 Hz (due t o a n e l a s t i c relaxation) may be calculated using the standard quantum-mechanical two-level system model (4) which y i e l d s

Q-1 = nPb2

2 u 7 ( 5 )

where

b

i s the root mean square couolinn constant of the two-level system enerqy s o l i t t i n g t o the longitudinal s t r a i n

i n

t h i s experiment, and p i s an e f f e c t i v e density of s t a t e s . Equation

5

c o r r e s ~ o n d s to the 'high temperature l i m i t ' and f o r

U Q,

1 HZ i s applicable f o r

T

2 1

mK.

The predicted i s frequency indenendent, in agreement with the present measurements from 0.005 - 50 Hz. This provides evid- ence t h a t the d i s t r i b u t i o n of relaxation times extends t o times longer than 100

S.

The observed internal f r i c t i o n i s temperature indenendent below

^Q,

4

K

a s oredicted by equation

5.

The r e s u l t s from both s i l i c a s y i e l d

Fb2 =

1.4 IQ7 J V 3 i n aood agreement with previous estimates

( 5 ) .

The second harmonic component can a r i s e from two d i f f e r e n t sources. Contribu- tions can come from e i t h e r the mechanical loss o r from the s t r a i n dependence of

y .

The contribution from the mechanical loss may be calculated using the same

C4- 1020 JOURNAL DE PHYSIQUE

d i s t r i b u t i o n o f r e l a x a t i o n t i n e s used t o o b t a i n equation 5, and q i v e s t h e major con- t r i b u t i o n t o

AT^.

The amplitude and phase o f t h e p r e d i c t e d second harmonic compon- e n t a r e i n agreenent w i t h t h a t observed.

The ' s t e p - f u n c t i o n ' s t r a i n r e s u l t s a r e a l s o exnlained u s i n g t h e same r e l a x a - t i o n time d i s t r i b u t i o n . The observed maonitude, s t r a i n dependence and l o g a r i t h m i c t i m e dependence o f AT a r e ~ r e d i c t e d . This time dependence comes about because two-level systems w i t 8 r e l a x a t i o n times dt

2

^T ?J .r0 c o n t r i b u t e t o AT,, whereas i n t h e s i n u s o i d a l case o n l y two-level systems w i t h ^T 2. l / w c o n t r i b u t e . The t h e o r y a l s o e x p l a i n s 14hy t h e maximum o f t h e symmetric temperature chanqe occurs a t a l a t e r time than t h a t o f t h e antisymmetric temperature chanqe (see F i o . I c and I d ) . T h i s e f f e c t a r i s e s because o f t h e delayed i r r e v e r s i b l e heat o u t p u t o f t h e two-level systems w i t h l o n g r e l a x a t i o n times ( T

2

0 . 1 ~ ~ ) .

I n summary, t h e measured energy d i s s i n a t i o n can be r e l a t e d t o e x i s t i n g h i q h e r temperature measurements, and understood on t h e b a s i s o f a standard model (4). T h i s model i s a general one assuming t h a t ' e l a s t i c d i p o l e s ' o r two-level systems w i t h a wide range o f r e l a x a t i o n times e x i s t i n amorphous s o l i d s , and so should a p p l y t o a wide range o f disordered m a t e r i a l s . The sign, magnitude and s t r a i n denendence o f t h e Grineisen parameter a r e more d i f f i c u l t t o understand i n q u a n t i t a t i v e terms.

These r e s u l t s depend on t h e i n t e r a c t i o n between two-level systems and t h e a p p l i e d s p r a i n i n a more d e t a i l e d way t h a n t h e d i s s i p a t i o n r e s u l t s , p r o b i n q t h e average c o u p l i n o t o s t r a i n e a r a t h e r than t h e mean square. Hoip~ever, i t i s - c l e a r t h a t these r e s u l t s f o r t h e Gruneisen parameter p r o v i d e an i m p o r t a n t c o n s t r a i n t on t h e p o s s i b l e microscopic models o f t h e d e f e c t s i n glasses.

Acknowledgements.- ble should l i k e t o thank Professor S i r B r i a n Pippard f o r many v a l u a b l e discussions. One o f us (O.B.W.) thanks t h e Science Research Council f o r support d u r i n g t h e course o f t h i s work.

References

(1) IIHITE, G.K., Phys. Rev. Lett.34 (1975) 205.

( 2 ) LYObl, K.G., SALINGER, G.L., Sl?E?IS3N, C.4., Phys. %v. B19 (1979) 4231.

(3) See e.g., GJeLQ0Y, K.S., PHILLIPS, W.A., P h i l . Ffao. t o b F p u b l i s h e d . ( 4 ) See e,u., JqCKLE, J.,Z. Physik ?57 (1972) 212.

(5) See e.g., GOLDING, S., G R A E B U E Q ~ ~ E . , SCHUTZ, 9.J., Phys. Rev. B E (1976) 1660.